Subgroups — C₂×#20289754 #28310215

Order 13568.

726 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719910	C₂	1	2	yes	maximal
2	#20289754	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719910	—	2	4	no	—
5	C₂²×#11719910	—	2	4	no	—
6	C₂²×#11719910	C₄	1	8	yes	complemented
7	C₂²×#11719910	C₂²	1	4	yes	—
8	C₂³×#88322	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	—	2	4	no	—
11	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
12	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
13	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
14	C₂×#11719910	—	2	16	no	—
15	C₂×#11719910	—	2	4	no	—
16	C₂×#11719910	—	2	4	no	—
17	C₂×#11719910	C₂×C₄	1	8	yes	complemented
18	C₂³×Dic₅₃	C₈	1	2	yes	—
19	C₂²×#88322	C₂×C₄	1	4	yes	—
20	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂²×#88322	D₄	1	4	yes	—
22	C₂²×#88322	D₄	1	2	yes	—
23	C₂²×#88322	Q₈	1	2	yes	—
24	C₂²×#88322	C₂³	1	1	yes	characteristic
25	C₂×C₄×Dic₅₃	—	2	2	no	—
26	C₂×C₄×Dic₅₃	—	2	2	no	—
27	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
28	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
29	C₈×Dic₅₃	—	2	4	no	—
30	C₈×Dic₅₃	—	2	4	no	—
31	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	—	2	2	no	—
33	C₂²×C₈×C₅₃	—	2	2	no	—
34	C₂²×C₈×C₅₃	C₈	1	4	yes	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
43	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	—	2	4	no	—
46	C₂²×Dic₅₃	—	2	4	no	—
47	C₂²×Dic₅₃	C₁₆	1	8	yes	complemented
48	C₂²×Dic₅₃	C₂×C₈	1	4	yes	—
49	C₂×C₈×C₅₃	—	2	8	no	—
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
53	C₄×Dic₅₃	—	2	4	no	—
54	C₄×Dic₅₃	—	2	2	no	—
55	C₄×Dic₅₃	—	2	2	no	—
56	C₂×#88322	C₄²	1	2	yes	—
57	C₂×#88322	C₂²⋊C₄	1	2	yes	—
58	C₂×#88322	C₂²⋊C₄	1	8	yes	—
59	C₂×#88322	C₄⋊C₄	1	2	yes	—
60	C₂×#88322	C₄⋊C₄	1	8	yes	—
61	C₂×#88322	C₂²×C₄	1	2	yes	—
62	C₂×#88322	C₂×D₄	1	1	yes	characteristic
63	C₂×#88322	C₂×D₄	1	2	yes	—
64	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
65	#11719910	—	2	8	no	—
66	C₂³×C₅₃	C₄×C₈	1	2	yes	—
67	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
68	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
69	C₂³×C₅₃	C₂×C₁₆	1	4	yes	—
70	C₂³×C₅₃	M₃₂	1	4	yes	—
71	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#95	1	4	yes	—
73	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
74	C₂×C₄×C₅₃	#97	1	4	yes	—
75	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
76	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
77	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
78	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
85	C₂×Dic₅₃	—	2	16	no	—
86	C₂×Dic₅₃	—	2	4	no	—
87	C₂×Dic₅₃	—	2	4	no	—
88	C₂×Dic₅₃	C₂×C₁₆	1	8	yes	complemented
89	C₈×C₅₃	—	2	4	no	—
90	#88322	#95	1	4	yes	—
91	#88322	C₂×C₂²⋊C₄	1	2	yes	—
92	#88322	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×#1374	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	#336	1	4	yes	—
95	C₂²×C₅₃	C₄×C₁₆	1	4	yes	—
96	C₂²×C₅₃	#346	1	4	yes	—
97	C₂²×C₅₃	C₂²⋊C₁₆	1	8	yes	—
98	C₂²×C₅₃	C₄⋊C₁₆	1	8	yes	—
99	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
100	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
101	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₂²×C₅₃	C₂²×C₁₆	1	2	yes	—
103	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
104	C₄×C₅₃	#336	1	4	yes	—
105	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
106	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
107	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
108	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
109	Dic₅₃	—	2	8	no	—
110	C₂³×C₁₆	—	53	2	no	—
111	C₂²×C₄×C₈	—	53	1	no	—
112	#1374	—	53	4	no	—
113	C₂×C₅₃	#1374	1	8	yes	complemented
114	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
115	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
116	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
117	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	2	yes	—
118	C₂×C₅₃	C₂×C₄⋊C₁₆	1	2	yes	—
119	C₂³×C₈	—	53	1	no	—
120	C₂³×C₈	#6157	1	1	yes	characteristic
121	C₂²×C₄²	—	53	1	no	—
122	C₂²×C₁₆	—	53	4	no	—
123	C₂²×C₁₆	—	53	8	no	—
124	C₂²×C₁₆	—	106	4	no	—
125	C₂²×C₁₆	—	106	4	no	—
126	C₂×C₄×C₈	—	53	2	no	—
127	C₂×C₄×C₈	—	53	2	no	—
128	C₂×C₄×C₈	—	106	4	no	—
129	C₅₃	C₂×#1374	1	1	yes	characteristic, complemented, Sylow
130	C₂³×C₄	—	53	2	no	—
131	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	53	4	no	—
135	C₂²×C₈	—	53	2	no	—
136	C₂²×C₈	—	53	2	no	—
137	C₂²×C₈	—	53	4	no	—
138	C₂²×C₈	—	53	1	no	—
139	C₂²×C₈	—	53	1	no	—
140	C₂²×C₈	#88324	1	4	yes	—
141	C₂²×C₈	C₂×#6157	1	2	yes	—
142	C₂×C₄²	—	53	2	no	—
143	C₂×C₄²	—	53	2	no	—
144	C₂×C₄²	—	106	2	no	—
145	C₂×C₄²	—	106	2	no	—
146	C₂×C₁₆	—	53	8	no	—
147	C₂×C₁₆	—	106	16	no	—
148	C₂×C₁₆	—	106	4	no	—
149	C₂×C₁₆	—	106	4	no	—
150	C₄×C₈	—	106	4	no	—
151	C₄×C₈	—	106	4	no	—
152	C₂⁴	D₅₃⋊C₈	1	1	yes	characteristic
153	C₂²×C₄	—	53	8	no	—
154	C₂²×C₄	—	53	4	no	—
155	C₂²×C₄	—	106	4	no	—
156	C₂²×C₄	—	106	4	no	—
157	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
158	C₂²×C₄	#11719936	1	2	yes	—
159	C₂²×C₄	C₄×#6157	1	2	yes	—
160	C₂²×C₄	C₄⋊#6157	1	2	yes	—
161	C₂²×C₄	C₂×#88324	1	2	yes	—
162	C₂²×C₄	#11719940	1	2	yes	—
163	C₂²×C₄	C₂²⋊#6157	1	2	yes	—
164	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
165	C₂×C₈	—	2	8	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	53	8	no	—
169	C₂×C₈	—	53	8	no	—
170	C₂×C₈	—	53	2	no	—
171	C₂×C₈	—	53	2	no	—
172	C₂×C₈	—	53	2	no	—
173	C₂×C₈	—	53	2	no	—
174	C₂×C₈	—	53	1	no	—
175	C₂×C₈	—	53	2	no	—
176	C₂×C₈	—	53	1	no	—
177	C₂×C₈	C₂×#88324	1	4	yes	—
178	C₄²	—	106	4	no	—
179	C₄²	—	106	2	no	—
180	C₄²	—	106	2	no	—
181	C₁₆	—	106	8	no	—
182	C₂³	C₄×#88324	1	2	yes	—
183	C₂³	#13133274	1	2	yes	—
184	C₂³	#13133284	1	2	yes	—
185	C₂³	D₅₃⋊C₁₆	1	4	yes	—
186	C₂³	#13133292	1	4	yes	—
187	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
188	C₂×C₄	—	53	8	no	—
189	C₂×C₄	—	106	4	no	—
190	C₂×C₄	—	106	4	no	—
191	C₂×C₄	—	106	16	no	—
192	C₂×C₄	(C₂×#6157)⋊C₄	1	4	yes	—
193	C₂×C₄	C₄×#88324	1	2	yes	—
194	C₂×C₄	#13133272	1	4	yes	—
195	C₂×C₄	C₂²⋊#88324	1	4	yes	—
196	C₂×C₄	#13133274	1	2	yes	—
197	C₂×C₄	C₄⋊#88324	1	4	yes	—
198	C₂×C₄	#13133284	1	2	yes	—
199	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
200	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
202	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
203	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
204	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
205	C₈	—	2	4	no	—
206	C₈	—	53	4	no	—
207	C₈	—	53	2	no	—
208	C₈	—	53	2	no	—
209	C₂²	#16811489	1	4	yes	—
210	C₂²	#16811509	1	4	yes	—
211	C₂²	#16811512	1	4	yes	—
212	C₂²	#16811518	1	8	yes	—
213	C₂²	#16811544	1	8	yes	—
214	C₂²	C₂×C₄×#88324	1	1	yes	characteristic
215	C₂²	C₂×#13133274	1	1	yes	characteristic
216	C₂²	C₂×#13133284	1	1	yes	characteristic
217	C₂²	C₂×D₅₃⋊C₁₆	1	2	yes	—
218	C₂²	C₂×#13133292	1	2	yes	—
219	C₄	—	106	8	no	—
220	C₄	#16811489	1	4	yes	—
221	C₄	C₂×(C₂×#6157)⋊C₄	1	1	yes	characteristic
222	C₄	C₂×#13133272	1	1	yes	characteristic
223	C₄	C₂×C₂²⋊#88324	1	1	yes	characteristic
224	C₄	C₂×C₄⋊#88324	1	1	yes	characteristic
225	C₂	#20289754	1	8	yes	direct product factor
226	C₂	C₂×#16811489	1	1	yes	characteristic
227	C₂	C₂×#16811509	1	1	yes	characteristic
228	C₂	C₂×#16811512	1	1	yes	characteristic
229	C₂	C₂×#16811518	1	2	yes	—
230	C₂	C₂×#16811544	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂×#6157
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂³×C₈×C₅₃	C₄
socle, prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×#88322	C₂³
hypercenter	C₂³×C₈	#6157
hypocenter	C₅₃	C₂×#1374
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #20289754	C₂	direct (N × H)
6. C₂²×#11719910	C₄	semidirect
17. C₂×#11719910	C₂×C₄	semidirect
47. C₂²×Dic₅₃	C₁₆	semidirect
88. C₂×Dic₅₃	C₂×C₁₆	semidirect
113. C₂×C₅₃	#1374	semidirect
129. C₅₃	C₂×#1374	semidirect
225. C₂	#20289754	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×#11719910	2
#20289754	2
C₂²×C₈×Dic₅₃	2
C₂×#1374	53

Subgroups — C2×#20289754 #28310215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20289754 #28310215